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Fvnctiones trigonometricae earvmqve delineationes

Svmmarivm

Haec lectio sinvm, cosinvm et tangentem vt fvnctiones variabilis realis, in radianis mensae, examinat. Earvm dominia, copias imaginvm, zera, valores extremos et paritatem definit, et demonstrat 2π2\pi periodvm positivam minimam sinvs et cosinvs esse, et π\pi tangentis. Delineationem sinvs constrvit circvmferentiam vnitariam evolvendo, delineationem cosinvs per translationem obtinet et asymptotos verticales tangentis tabvlis signorvm describit; inde delineationes secantis, cosecantis et cotangentis dedvcit. Areas comparando inaeqvalitatem sin⁡x<x<tan⁡x\sin x < x < \tan x pro 0<x<π20 < x < \frac{\pi}{2} demonstrat, et sinvsoides y=asin⁡(b(x−c))+dy = a \sin(b(x - c)) + d examinat, qvibvs phaenomena periodica describvntvr, velvt diei longitvdo, aestvs maris avt motvs rotae magnae.

Proposita discendi

  1. Sinvm, cosinvm et tangentem vt fvnctiones variabilis realis definire, et earvm copias imaginvm, zera, extrema, paritatem et periodos minimas demonstrare.
  2. Delineationes sex fvnctionvm trigonometricarvm ex circvmferentia vnitaria constrvere, asymptotos verticales tabvlis signorvm describendo.
  3. Inaeqvalitatem sin⁡x<x<tan⁡x\sin x < x < \tan x pro 0<x<π20 < x < \frac{\pi}{2} et marginem ∣sin⁡x∣≤∣x∣\abs{\sin x} \leq \abs{x} demonstrare, eisqve vti ad fvnctiones et aeqvationes examinandas.
  4. Amplitvdinem, periodvm, translationem phasis et translationem verticalem sinvsoidis vt transformationes delineationis sinvs interpretari, et sinvsoides phaenomenis periodicis accommodare.

Sinvs et cosinvs vt fvnctiones

In lectione de rationibvs trigonometricis sinvs et cosinvs angvli cvivslibet definiti svnt vt coordinatae pvncti P(θ)P(\theta) in qvo latvs terminale angvli, in positione normali, circvmferentiam vnitariam secat. Atqvi, secvndvm lectionem de angvlis eorvmqve mensvra, omnis nvmervs realis est mensvra in radianis alicvivs angvli orientati, et dvo angvli idem latvs terminale habent si et solvm si eorvm mensvrae mvltiplo integro ipsivs 2π2\pi differvnt. Qvod si ita est, sinvs et cosinvs considerari possvnt vt fvnctiones in toto R\mathbb{R} definitae: cviqve nvmero reali xx vna coordinata pvncti P(x)P(x) assignatvr.

Pro qvoqve nvmero reali xx, sit P(x)P(x) pvnctvm circvmferentiae vnitariae qvod angvlo xx radianorvm in positione normali respondet. Fvnctio sinvs ipsi xx ordinatam pvncti P(x)P(x) assignat, et fvnctio cosinvs eivs abscissam. Vtraqve dominivm habet copiam omnivm nvmerorvm realivm.
P(x)=(cos⁡x,sin⁡x),x∈RP(x) = (\cos x, \sin x), \quad x \in \mathbb{R}

Variabilis semper in radianis metitvr, non in gradibvs. Electio non est conventio indifferens: effecta capitis secvndi, velvt sin⁡x<x\sin x < x pro xx positivo, sinvm cvm ipsa variabili comparant, et haec comparatio sensvm habet solvm cvm xx est longitvdo arcvs in circvmferentia vnitaria percvrsi, qvod radianvs metitvr.

TheoremaProprietates sinvs et cosinvs

Copia imaginvm sinvs et cosinvs est intervallvm [−1,1]\icc{-1}{1}. Pro omni reali xx valent sin⁡(x+2π)=sin⁡x\sin(x + 2\pi) = \sin x et cos⁡(x+2π)=cos⁡x\cos(x + 2\pi) = \cos x; sinvs est fvnctio impar, cosinvs fvnctio par. Sinvs evanescit exacte in mvltiplis integris ipsivs π\pi, valet 11 exacte in nvmeris π2+2kπ\frac{\pi}{2} + 2k\pi et valet −1-1 exacte in nvmeris −π2+2kπ-\frac{\pi}{2} + 2k\pi, vbi kk integer est. Cosinvs evanescit exacte in nvmeris π2+kπ\frac{\pi}{2} + k\pi, valet 11 exacte in nvmeris 2kπ2k\pi et valet −1-1 exacte in nvmeris π+2kπ\pi + 2k\pi.

Omnia in circvmferentia vnitaria legvntvr, identitate Pythagorica et symmetriis in lectione de rationibvs trigonometricis demonstratis.

Demonstratio

  1. sin⁡2x=1−cos⁡2x≤1→−1≤sin⁡x≤1\sin^2 x = \resaltar{1 - \cos^2 x} \leq 1 \rightarrow \dato{c}{\resaltar{-1 \leq \sin x \leq 1}}

    Per identitatem Pythagoricam qvadratvm sinvs 11 non excedit, et sinvs inter −1-1 et 11 iacet; idem de cosinv valet.

  2. Q=(1−c2,c),(1−c2)2+c2=1→Rec⁡(sin⁡)=Rec⁡(cos⁡)=[−1,1]Q = \left( \sqrt{1 - c^2}, c \right), \quad \resaltar{\left( \sqrt{1 - c^2} \right)^2 + c^2} = \resaltar{1} \rightarrow \operatorname{Rec}(\sin) = \operatorname{Rec}(\cos) = \icc{-1}{1}

    Vicissim, dato cc cvm −1≤c≤1-1 \leq c \leq 1, pvnctvm QQ qvod seqvitvr in circvmferentia vnitaria iacet; per proprietatem operimenti axiomatis percvrsvs, in lectione de angvlis eorvmqve mensvra admissi, est pvnctvm P(x)P(x) alicvivs angvli xx, ideoqve sin⁡x=c\sin x = c. Eodem modo pvnctvm (c,1−c2)\left( c, \sqrt{1 - c^2} \right) ostendit cosinvm valorem cc accipere.

  3. sin⁡(x+2π)=sin⁡x,cos⁡(x+2π)=cos⁡x,sin⁡(−x)=−sin⁡x,cos⁡(−x)=cos⁡x\sin(x + 2\pi) = \sin x, \quad \cos(x + 2\pi) = \cos x, \quad \sin(-x) = -\sin x, \quad \cos(-x) = \cos x

    Angvli xx et x+2πx + 2\pi radianorvm idem latvs terminale habent, ita vt P(x+2π)=P(x)P(x + 2\pi) = P(x). Praeterea P(−x)P(-x) est reflexio ipsivs P(x)P(x) respectv axis horizontalis, vt in lectione de rationibvs trigonometricis demonstratvm est: sinvs impar est et cosinvs par.

  4. sin⁡x=0↔x=kπ,cos⁡x=0↔x=π2+kπ,k∈Z\sin x = 0 \leftrightarrow x = k\pi, \quad \cos x = 0 \leftrightarrow x = \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z}

    Sinvs evanescit cvm P(x)P(x) in axe horizontali iacet, id est cvm est (1,0)(1, 0) avt (−1,0)(-1, 0); haec svnt pvncta angvlorvm 00 et π\pi eorvmqve coterminalivm. Cosinvs evanescit cvm P(x)P(x) est (0,1)(0, 1) avt (0,−1)(0, -1).

  5. sin⁡x=1→cos⁡2x=1−1=0→P(x)=(0,1)\sin x = 1 \rightarrow \cos^2 x = \resaltar{1 - 1} = \resaltar{0} \rightarrow P(x) = (0, 1)

    Si sin⁡x=1\sin x = 1, identitas Pythagorica cogit cosinvm nvllvm esse, et pvnctvm P(x)P(x) est (0,1)(0, 1), qvod angvlo π2\frac{\pi}{2} respondet; si sin⁡x=−1\sin x = -1, pvnctvm est (0,−1)(0, -1), qvod ipsi −π2-\frac{\pi}{2} respondet.

  6. sin⁡x=1↔x=π2+2kπ,sin⁡x=−1↔x=−π2+2kπ,cos⁡x=1↔x=2kπ,cos⁡x=−1↔x=π+2kπ\sin x = 1 \leftrightarrow x = \frac{\pi}{2} + 2k\pi, \quad \sin x = -1 \leftrightarrow x = -\frac{\pi}{2} + 2k\pi, \quad \cos x = 1 \leftrightarrow x = 2k\pi, \quad \cos x = -1 \leftrightarrow x = \pi + 2k\pi

    Eadem ratione cosinvs valet 11 cvm P(x)=(1,0)P(x) = (1, 0) et valet −1-1 cvm P(x)=(−1,0)P(x) = (-1, 0). Svmmatim, valores extremi sinvs et cosinvs in nvmeris qvi seqvvntvr attingvntvr, vbi kk integer est.

Delineatio sinvs

Delineatio ipsivs y=sin⁡xy = \sin x obtinetvr circvmferentiam vnitariam svper axem horizontalem evolvendo: dvm pvnctvm P(x)P(x) circvmferentiam percvrrit, arcvs percvrsvs, longitvdinis xx, in axem abscissarvm transfertvr, et svper qvamqve abscissam altitvdo pvncti erigitvr. Symmetriae lectionis de rationibvs trigonometricis sinvnt totam delineationem ex prima eivs qvarta constrvere.

ExemplvmDelineatio sinvs vt circvmferentia evolvta

Figvra circvmferentiam vnitariam svper axem horizontalem evolvit, eadem vnitate in vtroqve: altitvdo pvncti P(x)P(x) ad abscissam xx transfertvr, et tota delineatio ex prima qvarta repetita et reflexa oritvr.

Demonstratio

  1. sin⁡0=0,sin⁡π2=1\sin 0 = 0, \quad \sin \frac{\pi}{2} = 1

    Dvm xx a 00 ad π2\frac{\pi}{2} progreditvr, pvnctvm P(x)P(x) primvm qvadrantem a (1,0)(1, 0) ad (0,1)(0, 1) percvrrit. Per proprietatem dimidiae conversionis axiomatis percvrsvs, eivs abscissa cos⁡x\cos x stricte decrescit a 11 ad 00; cvm eivs altitvdo sit radix non negativa 1−cos⁡2x\sqrt{1 - \cos^2 x}, stricte crescit a 00 ad 11. In lectione de fvnctionibvs trigonometricis inversis idem argvmentvm ad totvm intervallvm [−π2,π2]\icc{-\frac{\pi}{2}}{\frac{\pi}{2}} extenditvr.

  2. sin⁡π6=12,sin⁡π4=22,sin⁡π3=32\sin \frac{\pi}{6} = \frac{1}{2}, \quad \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}, \quad \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}

    Valores insignes in lectione de rationibvs trigonometricis compvtati tria pvncta media primi arcvs praebent.

  3. sin⁡(π−x)=sin⁡x\sin(\pi - x) = \sin x

    Inter π2\frac{\pi}{2} et π\pi, symmetria angvlorvm svpplementarivm dicit altitvdinem in π−x\pi - x eandem esse ac in xx: primvs arcvs symmetricvs est respectv rectae verticalis x=π2x = \frac{\pi}{2}, et ad 00 in π\pi descendit.

  4. sin⁡(x+π)=−sin⁡x\sin(x + \pi) = -\sin x

    Inter π\pi et 2π2\pi, pvnctvm P(x+π)P(x + \pi) est oppositvm pvncti P(x)P(x): altitvdo signvm mvtat, et delineatio arcvm praecedentem infra axem repetit, π\pi translatvm et reflexvm.

  5. sin⁡(x+2π)=sin⁡x,sin⁡(−x)=−sin⁡x\sin(x + 2\pi) = \sin x, \quad \sin(-x) = -\sin x

    Propter periodicitatem, portio inter 00 et 2π2\pi, qvae cyclvs appellatvr, dextrorsvm et sinistrorsvm repetitvr; et cvm sinvs impar sit, delineatio symmetrica est respectv originis.

Delineatio sinvs est vnda qvae inter rectas y=−1y = -1 et y=1y = 1 oscillat, axem horizontalem in mvltiplis ipsivs π\pi secat, et maxima in nvmeris π2+2kπ\frac{\pi}{2} + 2k\pi et minima in nvmeris −π2+2kπ-\frac{\pi}{2} + 2k\pi attingit, secvndvm theorema svperivs.

Delineatio praeterea dvas proprietates svpponit qvas circvmferentia evolvta svggerit, sed qvas instrvmenta hvivs partis demonstrare non sinvnt: delineationem esse cvrvam sine interrvptionibvs, et qvemqve arcvm svpra axem positvm semper deorsvm cvrvari. Vtraqve per limites definitvr et in cvrsv Calcvlvs I demonstratvr; hic admittvntvr. Nvllvm theorema hvivs partis ab eis pendet: in qvibvsdam tantvm problematis adhibentvr, qvae id expresse indicant.

Factvm admissvm (continvitas fvnctionvm trigonometricarvm earvmqve inversarvm). Sinvs et cosinvs in tota recta continvi svnt; tangens, secans, cosecans et cotangens in qvoqve intervallo in svo dominio contento; et arcvs sinvs, arcvs cosinvs et arcvs tangentis, qvi in lectione de fvnctionibvs trigonometricis inversis definientvr, in svis dominiis continvi svnt. Svmmae, prodvcta, qvotientes denominatore non nvllo et compositiones fvnctionvm continvarvm continvae svnt. Qvapropter factvm admissvm cvrsvs I de theoremate valoris intermedii, in lectione de fvnctionibvs rationalibvs envntiatvm, in omni intervallo clavso in dominio contento ad fvnctiones velvt sin⁡x−x10\sin x - \frac{x}{10}, tan⁡x−x\tan x - x avt arcsin⁡x+arctan⁡x\arcsin x + \arctan x adhibetvr: si in extremis signvm mvtant, in aliqvo pvncto interiore evanescvnt.
Factvm admissvm (concavitas sinvs). Sinvs stricte concavvs est in qvoqve intervallo [2kπ,(2k+1)π]\icc{2k\pi}{(2k + 1)\pi}, vbi kk integer est: si u<x<vu < x < v svnt pvncta hvivs intervalli, pvnctvm delineationis abscissae xx stricte svpra segmentvm iacet qvod pvncta delineationis abscissarvm uu et vv ivngit.
u<x<v→sin⁡x>(v−x)sin⁡u+(x−u)sin⁡vv−uu < x < v \rightarrow \sin x > \frac{(v - x) \sin u + (x - u) \sin v}{v - u}

Fvnctiones periodicae

Fvnctio ff periodica est si exstat nvmervs p>0p > 0 talis vt, pro omni xx dominii, x+px + p et x−px - p in dominio sint et valeat f(x+p)=f(x)f(x + p) = f(x); talis nvmervs periodvs ipsivs ff appellatvr. Si pp periodvs est, etiam 2p2p, 3p3p et, per indvctionem, omnia eivs mvltipla natvralia periodi svnt: f(x+2p)=f((x+p)+p)=f(x+p)=f(x)f(x + 2p) = f((x + p) + p) = f(x + p) = f(x). Fvnctio periodica igitvr infinitas periodos habet, et maxime notabilis est minima earvm, cvm exstat, qvae periodvs minima appellatvr. Eivs delineatio obtinetvr portionem intervallo longitvdinis periodo aeqvalis respondentem itervm atqve itervm repetendo.

LemmaPeriodi svnt mvltipla periodi minimae

Si qq est periodvs minima fvnctionis ff, periodi ipsivs ff svnt exacte nvmeri kqkq, vbi kk natvralis est.

Periodvs qvaelibet per qq dividitvr, vt in divisione cvm residvo, et ostenditvr residvvm positivvm esse non posse. Figvra fvnctionem periodi minimae qq adhibet.

Demonstratio

  1. f(x+kq)=f(x),k∈N\dato{m}{f(x + kq) = f(x)}, \quad k \in \mathbb{N}

    Qvodqve mvltiplvm natvrale ipsivs qq periodvs est, vt modo per indvctionem visvm est.

  2. kq≤p<(k+1)q→0≤r=p−kq<q\resaltar{kq \leq p < (k + 1)q} \rightarrow \dato{r}{\resaltar{0 \leq r = p - kq < q}}

    Vicissim, sit pp periodvs; propter minimalitatem ipsivs qq, est p≥qp \geq q. Per proprietatem Archimedeam nvmerorvm realivm, in lectione cvrsvs I de expressione decimali nvmerorvm rationalivm demonstratam, et principivm bonae ordinationis, vt in theoremate redvctionis ad primam conversionem lectionis de angvlis eorvmqve mensvra, exstat natvralis k≥1k \geq 1 cvm kq≤p<(k+1)qkq \leq p < (k + 1)q, et residvvm r=p−kqr = p - kq inter 00 et qq iacet, hoc exclvso.

  3. f(x+r)=f(x+r+kq)=f(x+p)=f(x)f(x + r) = f(x + \resaltar{r + kq}) = f(x + \resaltar{p}) = f(x)

    Si rr positivvm esset, periodvs esset: progrediendo rr et deinde kqkq progreditvr pp, et tam kqkq qvam pp periodi svnt.

  4. r=0→p=kqr = 0 \rightarrow p = kq

    Sed periodvs positiva minor qvam qq minimalitati ipsivs qq repvgnat. Vnde residvvm nvllvm est, et pp est mvltiplvm natvrale ipsivs qq.

TheoremaPeriodvs minima sinvs et cosinvs

Nvmervs 2π2\pi est periodvs minima sinvs et cosinvs.

Per theorema proprietatvm, 2π2\pi est periodvs vtrivsqve fvnctionis. Restat vt ostendatvr nvllvm nvmervm inter 00 et 2π2\pi periodvm esse, et ad hoc satis est inspicere vbi maximvm attingatvr.

Demonstratio

  1. sin⁡(π2+p)=sin⁡π2=1\dato{m}{\sin \left( \frac{\pi}{2} + p \right) = \sin \frac{\pi}{2} = 1}

    Svpponamvs pp, cvm 0<p<2π0 < p < 2\pi, periodvm sinvs esse. Aeqvalitatem sin⁡(x+p)=sin⁡x\sin(x + p) = \sin x in x=π2x = \frac{\pi}{2} compvtando, nvmervs π2+p\frac{\pi}{2} + p etiam pvnctvm maximi esset.

  2. π2+p=π2+2kπ→p=2kπ\frac{\pi}{2} + p = \frac{\pi}{2} + 2k\pi \rightarrow p = 2k\pi

    Per theorema proprietatvm, sinvs valet 11 solvm in nvmeris π2+2kπ\frac{\pi}{2} + 2k\pi: ergo pp esset mvltiplvm integrvm ipsivs 2π2\pi, qvod inter 00 et 2π2\pi fieri non potest. Aliis verbis, dvo maxima consecvtiva sinvs exacte 2π2\pi distant.

  3. cos⁡p=cos⁡0=1→p=2kπ\cos p = \cos 0 = 1 \rightarrow p = 2k\pi

    Pro cosinv similiter in x=0x = 0 ratiocinamvr: periodvs pp daret cos⁡p=cos⁡0=1\cos p = \cos 0 = 1, et cosinvs valet 11 solvm in mvltiplis integris ipsivs 2π2\pi.

Per lemma svperivs, periodi sinvs et cosinvs svnt exacte nvmeri 2kπ2k\pi, vbi kk natvralis est.

Cosinvs vt sinvs translatvs

Delineatio cosinvs novam constrvctionem non reqvirit, qvia est delineatio sinvs translata.

TheoremaCosinvs est sinvs translatvs

Pro omni reali xx valet cos⁡x=sin⁡(x+π2)\cos x = \sin \left( x + \frac{\pi}{2} \right). Qvapropter delineatio ipsivs y=cos⁡xy = \cos x est delineatio sinvs π2\frac{\pi}{2} vnitates sinistrorsvm translata.

Symmetriae angvlorvm complementarivm et svpplementarivm conivngvntvr, ac deinde theorema transformationvm lectionis cvrsvs I de operationibvs fvnctionvm adhibetvr.

Demonstratio

  1. cos⁡x=sin⁡(π2−x)\cos x = \dato{c}{\sin \left( \frac{\pi}{2} - x \right)}

    Per symmetriam angvlorvm complementarivm, cosinvs ipsivs xx est sinvs ipsivs π2−x\frac{\pi}{2} - x.

  2. cos⁡x=sin⁡(π−(π2−x))=sin⁡(x+π2)\cos x = \sin \left( \resaltar{\pi - \left( \frac{\pi}{2} - x \right)} \right) = \dato{t}{\sin \left( \resaltar{x + \frac{\pi}{2}} \right)}

    Per symmetriam angvlorvm svpplementarivm, sin⁡(π−u)=sin⁡u\sin(\pi - u) = \sin u; cvm u=π2−xu = \frac{\pi}{2} - x, angvlvs π−u\pi - u est π2+x\frac{\pi}{2} + x.

  3. y=sin⁡(x−(−π2))y = \sin \left( x - \left( -\frac{\pi}{2} \right) \right)

    Per theorema transformationvm, delineatio ipsivs y=f(x−h)y = f(x - h) est delineatio ipsivs ff hh vnitates horizontaliter translata. Hic ff est sinvs et h=−π2h = -\frac{\pi}{2}: qvodqve pvnctvm delineationis sinvs π2\frac{\pi}{2} sinistrorsvm movetvr. Maximvm ipsivs (π2,1)\left( \frac{\pi}{2}, 1 \right) ad (0,1)(0, 1) transit.

  4. cos⁡(−x)=cos⁡x,cos⁡0=1,cos⁡π=−1\cos(-x) = \cos x, \quad \cos 0 = 1, \quad \cos \pi = -1

    In delineatione translata proprietates cosinvs legvntvr: symmetrica est respectv axis verticalis, qvia cosinvs par est, axem horizontalem in nvmeris π2+kπ\frac{\pi}{2} + k\pi secat, et maximvm 11 in mvltiplis paribvs ipsivs π\pi attingit, minimvm −1-1 in imparibvs.